= Electrical reactance bench measurement
{title2=$X=|Z|\sin\phi$}
The motivating observation is a pair of sinusoids on an oscilloscope: changing <frequency> changes both their relative <amplitudes> and their <phase difference>. Gary Tuttle's EE 201 laboratory uses resistor voltage as the current-phase reference, compares predictions with readings at several <frequencies>, and explicitly asks students to measure component values and account for coil <electrical resistance>. The following bench arrangement applies that method to a <series RLC circuit>; its component values and illustrated traces are our own toy example. https://gtuttle.net/circuits/lab/ac_circuits_1.pdf[Gary Tuttle, EE 201 Lab 9: AC circuits, especially the notes and RC/RL measurements]{ref}.
\Image[/reactance-bench-phase-sensing.png]
{title=Toy AC bench arrangement and predicted voltage/current timing}
{description=Top: a sine generator drives ideal $L=10\,\mathrm{mH}$, $C=1\,\mu\mathrm F$, $R=100\,\Omega$, and current-sensing $R_s=10\,\Omega$ in series. Channel A measures loaded terminal voltage; channel B measures $v_s=R_si$ relative to the common return. Bottom: calculated separate voltage traces at $f=500\,\mathrm{Hz}$ and terminal $V_{\rm rms}=1.00\,\mathrm V$, not an oscilloscope recording or measured data. The rising zero of channel B occurs $0.3835\,\mathrm{ms}$ earlier: $\phi=-69.02^\circ$ and predicted $I_{\rm rms}=3.255\,\mathrm{mA}$. The two trace axes have different voltage scales.}
Connect a low-voltage sine generator to a series chain ending in a known sensing <resistor> $R_s$ at the common return. Use a <resistor> whose unwanted <inductance> is negligible over the chosen <frequency> range. An oscilloscope displays voltage against time: channel A reads the loaded voltage across the whole chain, and channel B reads the voltage $v_s$ across $R_s$. Both channel ground connections and the generator return meet at that same node. Ordinary bench oscilloscope grounds are electrically common; attaching one to an intermediate circuit node would change the circuit. A separate floating component voltage needs a suitable differential measurement, rather than moving a ground clip. https://www.tek.com/en/documents/technical-brief/floating-oscilloscope-measurements-and-operator-protection[Tektronix, Floating Oscilloscope Measurements and Operator Protection: ground-referenced inputs]{ref}.
After the free transient decays, record <frequency> $f$, period $T=1/f$, both voltage <amplitudes>, and matched rising zero-crossing times $t_A,t_B$. Check that the traces are sinusoidal. Define $\Delta t=t_B-t_A$ on the branch $|\Delta t|<T/2$. With negligible probe loading and negligible sensing-<resistor> <electrical reactance>, $v_s=R_si$, so channel B is the <electric current> reference, and in the <reactance phase convention>
$$
I_{\rm rms}=\frac{V_{s,\rm rms}}{R_s},\qquad
|Z_{\rm chain}|=\frac{V_{A,\rm rms}}{I_{\rm rms}},\qquad
\phi=\frac{2\pi\Delta t}{T},\qquad
X_{\rm chain}=|Z_{\rm chain}|\sin\phi,\qquad
R_{\rm chain}=|Z_{\rm chain}|\cos\phi.
$$
A later current crossing gives $\Delta t>0$, hence positive <electrical reactance>; an earlier crossing gives negative <electrical reactance>. Use the same <amplitude> convention for both channels. An amplitude-only ratio recovers $|Z|$, so it cannot distinguish the two signs: timing supplies the missing information. The formulas follow from $Z=|Z|e^{\mathrm i\phi}$ and the chosen terminal orientation.
Begin with a known <resistor> replacing the reactive chain: its expected <phase difference> is approximately zero, testing channel polarity and relative delay. Measure the same signal with both channels to establish the instrumental delay baseline, then exchange probes to check that a phase offset does not belong to one channel. Next test an <inductor> and a <capacitor> separately with $R_s$: after subtracting $R_s$ from the inferred real part, expect approximately $X/f=2\pi L$ and $fX=-1/(2\pi C)$, respectively. Finally combine $L,C,R,R_s$ and sweep through the predicted <resonance>. These are proposed controls and consequences of the model, not reported observations.
For the toy <series RLC circuit>, the prediction at $500\,\mathrm{Hz}$ is $X=-286.894\,\Omega$, $|Z|=307.259\,\Omega$, $\phi=-69.02^\circ$, and $V_{s,\rm rms}=32.55\,\mathrm{mV}$. At $5000\,\mathrm{Hz}$, $X=+282.328\,\Omega$ and $\phi=+68.71^\circ$. Near $1591.55\,\mathrm{Hz}$ the model predicts zero <phase difference> and $V_{s,\rm rms}=90.91\,\mathrm{mV}$. Keep actual loaded $V_{A,\rm rms}$ fixed or normalize by it at each <frequency>; the generator's display alone does not establish that voltage. The $10\,\Omega$ sensing <resistor> is part of the $110\,\Omega$ total and therefore changes the predicted peak <electric current>.
Real components add unintended <electrical impedance>: coil wire has <electrical resistance>, nearby turns add <capacitance>, and <capacitor> leads add <inductance>. These effects can produce a component's own <resonance>, beyond which the ideal $\omega L$ or $-1/(\omega C)$ approximation fails. Work in a range where that approximation is adequate, measure coil wire <electrical resistance>, and compare the inferred $R_{\rm chain}$ with the sensing <resistor>, intended <resistor>, and component losses. A measured nonzero real part is evidence of loss in the assembly, not negative or positive <electrical reactance> becoming heat. https://www.keysight.com.cn/cn/zh/assets/7018-06840/application-notes/5950-3000.pdf[Keysight, Impedance Measurement Handbook, §§1.5.1 and 5.2.1: component frequency response and parasitic elements]{ref}; https://gtuttle.net/circuits/lab/ac_circuits_1.pdf[Tuttle, EE 201 Lab 9: parasitic coil series resistance]{ref}.
The probes also add <electrical resistance> and <capacitance> to the measured nodes; their loading becomes more important as <frequency> increases. Use compensated probes with known input <electrical impedance> and short ground connections, and repeat with a smaller drive voltage to test whether the inferred $X$ depends on <amplitude>. https://www.tek.com/en/documents/application-note/how-oscilloscope-probes-affect-your-measurement[Tektronix, How Oscilloscope Probes Affect Your Measurement: resistive and capacitive loading]{ref}. Record actual voltage and phase readings separately from calculated predictions, with component tolerances and timing uncertainty. A phase error $\delta\phi$ alone produces approximately $\delta X=|Z|\cos\phi\,\delta\phi$, so near zero <electrical reactance>, a small delay can imitate a small residual $X$. Agreement within stated uncertainties tests the model; the smooth toy curves establish no experimental accuracy.
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